How to make eyelids that close with a shape key?
If you made a second shape key to "fix" it, you can use a Blend Key without the need to manipulate multiple keys.

In the shapekey dropdown, next to the min/max setting, deselect "Base" and select the key you want to blend with the current key.

Imagine it like parenting key1 to key2, it will blend the two shapes when you use Key2.

The problem in your case seems to be that the eyelid has very little geometry compared to the eye, so the shape key interpolates it linearly along the edges which intersect the eye.

A fix would be to increase geometry to adjust it to the eye and it should work fine if it has more steps in-between.